Steenrod problem

In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds.[1]

Formulation

Let ${\displaystyle M}$ be a closed, oriented manifold of dimension ${\displaystyle n}$, and let ${\displaystyle [M]\in H_{n}(M)}$ be its orientation class. Here ${\displaystyle H_{n}(M)}$ denotes the integral, ${\displaystyle n}$-dimensional homology group of ${\displaystyle M}$. Any continuous map ${\displaystyle f\colon M\to X}$ defines an induced homomorphism ${\displaystyle f_{*}\colon H_{n}(M)\to H_{n}(X)}$.[2] A homology class of ${\displaystyle H_{n}(X)}$ is called realisable if it is of the form ${\displaystyle f_{*}[M]}$ where ${\displaystyle [M]\in H_{n}(M)}$. The Steenrod problem is concerned with describing the realisable homology classes of ${\displaystyle H_{n}(X)}$.[3]

Results

All elements of ${\displaystyle H_{k}(X)}$ are realisable by smooth manifolds provided ${\displaystyle k\leq 6}$. Any elements of ${\displaystyle H_{n}(X)}$ are realisable by a mapping of a Poincaré complex provided ${\displaystyle n\neq 3}$. Moreover, any cycle can be realized by the mapping of a pseudo-manifold.[3]

The assumption that M be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of ${\displaystyle H_{n}(X,\mathbb {Z} _{2})}$, where ${\displaystyle \mathbb {Z} _{2}}$ denotes the integers modulo 2, can be realized by a non-oriented manifold, ${\displaystyle f\colon M^{n}\to X}$.[3]

Conclusions

For smooth manifolds M the problem reduces to finding the form of the homomorphism ${\displaystyle \Omega _{n}(X)\to H_{n}(X)}$, where ${\displaystyle \Omega _{n}(X)}$ is the oriented bordism group of X.[4] The connection between the bordism groups ${\displaystyle \Omega _{*}}$ and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms ${\displaystyle H_{*}(\operatorname {MSO} (k))\to H_{*}(X)}$.[3][5] In his landmark paper from 1954,[5] René Thom produced an example of a non-realisable class, ${\displaystyle [M]\in H_{7}(X)}$, where M is the Eilenberg–MacLane space ${\displaystyle K(\mathbb {Z} _{3}\oplus \mathbb {Z} _{3},1)}$.

See also

• Singular homology
• Pontryagin-Thom construction
• Cobordism

References

1. Eilenberg, Samuel (1949). "On the problems of topology". Annals of Mathematics. 50: 247–260. doi:10.2307/1969448.
2. Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
3. Encyclopedia of Mathematics. "Steenrod Problem". Retrieved October 29, 2020.
4. Rudyak, Yuli B. (1987). "Realization of homology classes of PL-manifolds with singularities". Mathematical Notes. 41 (5): 417–421. doi:10.1007/bf01159869.
5. Thom, René (1954). "Quelques propriétés globales des variétés differentiable". Commentarii Mathematici Helvetici (in French). 28: 17–86. doi:10.1007/bf02566923.

External links

• Thom construction and the Steenrod problem on MathOverflow
• Explanation for the Pontryagin-Thom construction